1,236 research outputs found

### A Shape Theorem for Riemannian First-Passage Percolation

Riemannian first-passage percolation (FPP) is a continuum model, with a
distance function arising from a random Riemannian metric in $\R^d$. Our main
result is a shape theorem for this model, which says that large balls under
this metric converge to a deterministic shape under rescaling. As a
consequence, we show that smooth random Riemannian metrics are geodesically
complete with probability one

### Coexistence for a multitype contact process with seasons

We introduce a multitype contact process with temporal heterogeneity
involving two species competing for space on the $d$-dimensional integer
lattice. Time is divided into seasons called alternately season 1 and season 2.
We prove that there is an open set of the parameters for which both species can
coexist when their dispersal range is large enough. Numerical simulations also
suggest that three species can coexist in the presence of two seasons. This
contrasts with the long-term behavior of the time-homogeneous multitype contact
process for which the species with the higher birth rate outcompetes the other
species when the death rates are equal.Comment: Published in at http://dx.doi.org/10.1214/09-AAP599 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### A Differentiation Theory for It\^o's Calculus

A peculiar feature of It\^o's calculus is that it is an integral calculus
that gives no explicit derivative with a systematic differentiation theory
counterpart, as in elementary calculus. So, can we define a pathwise stochastic
derivative of semimartingales with respect to Brownian motion that leads to a
differentiation theory counterpart to It\^o's integral calculus? From It\^o's
definition of his integral, such a derivative must be based on the quadratic
covariation process. We give such a derivative in this note and we show that it
leads to a fundamental theorem of stochastic calculus, a generalized stochastic
chain rule that includes the case of convex functions acting on continuous
semimartingales, and the stochastic mean value and Rolle's theorems. In
addition, it interacts with basic algebraic operations on semimartingales
similarly to the way the deterministic derivative does on deterministic
functions, making it natural for computations. Such a differentiation theory
leads to many interesting applications some of which we address in an upcoming
article.Comment: 10 pages, 9/9 papers from my 2000-2006 collection. I proved these
results and more earlier in 2004. I generalize this theory in upcoming
articles. I also apply this theory in upcoming article

### Jamming Percolation and Glass Transitions in Lattice Models

A new class of lattice gas models with trivial interactions but constrained
dynamics are introduced. These are proven to exhibit a dynamical glass
transition: above a critical density, rho_c, ergodicity is broken due to the
appearance of an infinite spanning cluster of jammed particles. The fraction of
jammed particles is discontinuous at the transition, while in the unjammed
phase dynamical correlation lengths and timescales diverge as
exp[C(rho_c-rho)^(-mu)]. Dynamic correlations display two-step relaxation
similar to glass-formers and jamming systems.Comment: 4 pages, 2 figs. Final version accepted for publication in Phys. Rev.
Let

### Lightning -Apollo To Shuttle

The lightning discharge that struck the Apollo 12 spacecraft thirty-six seconds after launch pointed up a whole series of problems that called out for answers if the Manned Space Program were to proceed with minimum impact to future missions and the crews that would fly them. This paper traces the history of lightning study by the Kennedy Space Center from then to now with particular emphasis on the potential problems that may arise in the process of getting ready for and carrying out the Space Shuttle Program

### Affleck-Kennedy-Lieb-Tasaki State on a Honeycomb Lattice is a Universal Quantum Computational Resource

Universal quantum computation can be achieved by simply performing
single-qubit measurements on a highly entangled resource state, such as cluster
states. The family of Affleck-Kennedy-Lieb-Tasaki states has recently been
intensively explored and shown to provide restricted computation. Here, we show
that the two-dimensional Affleck-Kennedy-Lieb-Tasaki state on a honeycomb
lattice is a universal resource for measurement-based quantum computation.Comment: 4+2 pages, 4 figures, PRL short version of arXiv:1009.2840, see also
alternative approach by A. Miyake, arXiv:1009.349

### Phase Transition with the Berezinskii--Kosterlitz--Thouless Singularity in the Ising Model on a Growing Network

We consider the ferromagnetic Ising model on a highly inhomogeneous network
created by a growth process. We find that the phase transition in this system
is characterised by the Berezinskii--Kosterlitz--Thouless singularity, although
critical fluctuations are absent, and the mean-field description is exact.
Below this infinite order transition, the magnetization behaves as
$exp(-const/\sqrt{T_c-T})$. We show that the critical point separates the phase
with the power-law distribution of the linear response to a local field and the
phase where this distribution rapidly decreases. We suggest that this phase
transition occurs in a wide range of cooperative models with a strong
infinite-range inhomogeneity. {\em Note added}.--After this paper had been
published, we have learnt that the infinite order phase transition in the
effective model we arrived at was discovered by O. Costin, R.D. Costin and C.P.
Grunfeld in 1990. This phase transition was considered in the papers: [1] O.
Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531 (1990); [2] O.
Costin and R.D. Costin, J. Stat. Phys. 64, 193 (1991); [3] M. Bundaru and C.P.
Grunfeld, J. Phys. A 32, 875 (1999); [4] S. Romano, Mod. Phys. Lett. B 9, 1447
(1995). We would like to note that Costin, Costin and Grunfeld treated this
model as a one-dimensional inhomogeneous system. We have arrived at the same
model as a one-replica ansatz for a random growing network where expected to
find a phase transition of this sort based on earlier results for random
networks (see the text). We have also obtained the distribution of the linear
response to a local field, which characterises correlations in this system. We
thank O. Costin and S. Romano for indicating these publications of 90s.Comment: 5 pages, 2 figures. We have added a note indicating that the infinite
order phase transition in the effective model we arrived at was discovered in
the work: O. Costin, R.D. Costin and C.P. Grunfeld, J. Stat. Phys. 59, 1531
(1990). Appropriate references to the papers of 90s have been adde

### On the push&pull protocol for rumour spreading

The asynchronous push&pull protocol, a randomized distributed algorithm for
spreading a rumour in a graph $G$, works as follows. Independent Poisson clocks
of rate 1 are associated with the vertices of $G$. Initially, one vertex of $G$
knows the rumour. Whenever the clock of a vertex $x$ rings, it calls a random
neighbour $y$: if $x$ knows the rumour and $y$ does not, then $x$ tells $y$ the
rumour (a push operation), and if $x$ does not know the rumour and $y$ knows
it, $y$ tells $x$ the rumour (a pull operation). The average spread time of $G$
is the expected time it takes for all vertices to know the rumour, and the
guaranteed spread time of $G$ is the smallest time $t$ such that with
probability at least $1-1/n$, after time $t$ all vertices know the rumour. The
synchronous variant of this protocol, in which each clock rings precisely at
times $1,2,\dots$, has been studied extensively. We prove the following results
for any $n$-vertex graph: In either version, the average spread time is at most
linear even if only the pull operation is used, and the guaranteed spread time
is within a logarithmic factor of the average spread time, so it is $O(n\log
n)$. In the asynchronous version, both the average and guaranteed spread times
are $\Omega(\log n)$. We give examples of graphs illustrating that these bounds
are best possible up to constant factors. We also prove theoretical
relationships between the guaranteed spread times in the two versions. Firstly,
in all graphs the guaranteed spread time in the asynchronous version is within
an $O(\log n)$ factor of that in the synchronous version, and this is tight.
Next, we find examples of graphs whose asynchronous spread times are
logarithmic, but the synchronous versions are polynomially large. Finally, we
show for any graph that the ratio of the synchronous spread time to the
asynchronous spread time is $O(n^{2/3})$.Comment: 25 page

### The 2D AKLT state on the honeycomb lattice is a universal resource for quantum computation

Universal quantum computation can be achieved by simply performing
single-qubit measurements on a highly entangled resource state. Resource states
can arise from ground states of carefully designed two-body interacting
Hamiltonians. This opens up an appealing possibility of creating them by
cooling. The family of Affleck-Kennedy-Lieb-Tasaki (AKLT) states are the ground
states of particularly simple Hamiltonians with high symmetry, and their
potential use in quantum computation gives rise to a new research direction.
Expanding on our prior work [T.-C. Wei, I. Affleck, and R. Raussendorf, Phys.
Rev. Lett. 106, 070501 (2011)], we give detailed analysis to explain why the
spin-3/2 AKLT state on a two-dimensional honeycomb lattice is a universal
resource for measurement-based quantum computation. Along the way, we also
provide an alternative proof that the 1D spin-1 AKLT state can be used to
simulate arbitrary one-qubit unitary gates. Moreover, we connect the quantum
computational universality of 2D random graph states to their percolation
property and show that these states whose graphs are in the supercritical (i.e.
percolated) phase are also universal resources for measurement-based quantum
computation.Comment: 21 pages, 13 figures, long version of Phys. Rev. Lett. 106, 070501
(2011) or arXiv:1102.506

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